This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353508 #14 May 17 2022 19:51:19
%S A353508 1,0,0,0,1,0,2,0,2,1,4,0,8,2,11,4,21,5,37,12,57,25,104,38,177,79,292,
%T A353508 149,513,251,876,482,1478,871,2562,1533,4387,2815,7473,5036,12908,
%U A353508 8935,22135,16085,37940,28611,65422,50731,112459,90408,193386,160119,333513
%N A353508 Number of integer compositions of n with no ones or runs of length 1.
%H A353508 Alois P. Heinz, <a href="/A353508/b353508.txt">Table of n, a(n) for n = 0..1000</a>
%e A353508 The a(0) = 1 through a(14) = 11 compositions (empty columns indicated by dots, 0 is the empty composition):
%e A353508 0 . . . 22 . 33 . 44 333 55 . 66 22333 77
%e A353508 222 2222 2233 444 33322 2255
%e A353508 3322 2244 3344
%e A353508 22222 3333 4433
%e A353508 4422 5522
%e A353508 22233 22244
%e A353508 33222 44222
%e A353508 222222 222233
%e A353508 223322
%e A353508 332222
%e A353508 2222222
%p A353508 b:= proc(n,h) option remember; `if`(n=0, 1, add(
%p A353508 `if`(i<>h, add(b(n-i*j, i), j=2..n/i), 0), i=2..n/2))
%p A353508 end:
%p A353508 a:= n-> b(n, 0):
%p A353508 seq(a(n), n=0..60); # _Alois P. Heinz_, May 17 2022
%t A353508 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1]&&!MemberQ[Length/@Split[#],1]&]],{n,0,15}]
%Y A353508 The version for partitions is A339222.
%Y A353508 Compositions counted by their run-lengths:
%Y A353508 - For run-lengths <= 1 we have A003242, ranked by A333489.
%Y A353508 - For run-lengths = 2 we have A003242 aerated.
%Y A353508 - For run-lengths > 1 we have A114901, ranked by A353427.
%Y A353508 - For run-lengths <= 2 we have A128695 matching A335464.
%Y A353508 - For run-lengths > 2 we have A353400, partitions A100405.
%Y A353508 - For run-lengths all prime we have A353401.
%Y A353508 - For run-lengths and parts > 2 we have A353428.
%Y A353508 A008466 counts compositions with some part > 2.
%Y A353508 A011782 counts compositions.
%Y A353508 A106356 counts compositions by number of adjacent equal parts.
%Y A353508 A261983 counts non-anti-run compositions.
%Y A353508 A274174 counts compositions with equal parts contiguous.
%Y A353508 Cf. A005811, A078012, A238279, A329739, A333755, A351013, A353390.
%K A353508 nonn
%O A353508 0,7
%A A353508 _Gus Wiseman_, May 17 2022
%E A353508 a(41)-a(52) from _Alois P. Heinz_, May 17 2022